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IWA Publishing 2013

Journai of iHydroinformatics | 1 5 1 | 2013

Automated parameter optimization of

water distribution

system

Maikel Mndez, Jos A . Araya and Luis D. Snchez

ABSTRACT

The hydraulic model EPANET was applied and calibrated for the water distribution system (WDS) of La

Sirena, Colombia. The Parameter ESTimator (PEST) was used for parameter optimization and

sensitivity analysis. Observation data included levels at water storage tanks and pressures at

monitoring nodes. Adjustable parameters were grouped into different classes according to two

different scenarios identified as constrained and unconstrained. These scenarios were established to

evaluate the effect of parameter space size and compensating errors over the calibration process.

Results from the unconstrained scenario, where 723 adjustable parameters were declared, showed

that considerable compen sating errors are introduced into the optimization process if all parameters

wer e open to ad justment. The constrained scenario on the other h and, represented a more properly

discretized scheme as parameters were grouped into classes of similar characteristics and

Insensitive parameters w ere fixed . This had a profound impact on th e param eter space as adjustable

parameters we re reduced to 24. The constrained solution, even whe n it is valid only for the system s

normal operating conditions, clearly demonstrates that Parallel PEST (PPEST) has the potential to be

used in the calibration of WDS models. Nevertheless, further investigation is needed to determine

PPEST s perfo rman ce in complex WDS models.

Key words

| calibration, EPANET, model, optimization, parameter, sensitivity

iVlaikel Mndez

(corresponding author)

Jos A Araya

Centro de Investigaciones en Vivienda y

construccin (CIVCO), Escuela de ingeniera en

Construccin,

Instituto Tecnolgico de Costa Rica,

APDO 159-7050,

Cartago,

Costa Rica

E-mail:

mamendez@itcr ac cr

Luis D S nciiez

Instituto Cinara,

universidad del Valle - Facultad de Ingeniera,

Cali,

Colombia

INTRODUCTION

W ater distribution system (WDS) models canbeused foravar-

iety of purposes including design, managem ent, m aintenance,

planning and scenario studies. Nevertheless, in order to be

reliable

a model must adequately predict the behavior of the

actual system under a wide range of conditions and for an

extended period of time (Machell

et al

2010). This can b e

accom plished by calibrating th e m odel using a set of field

measurements or observations, mainly water storage tanit

(WST) levels, nodal pressures and flow rates. The calibration

ofa W Smodeliscarried out by optimizingthevalues of

phys-

ical and conceptual parameters involved in the model,

including pipe roughness-coefficients, minor losses, demand

pattern factors, nodal demands, control valves and pump

characteristics (USEPA2005; Koppel Vassiljevaoog).

Calibration is also referred to as an inverse problem,

since observed values are quantitatively compared to

doi: 10.2166/hydro.2012.028

model predictions until the optimum set of parameters is

found (Gallagher Doh erty 2007). A mo del is considered

to be calibrated for one set of operating conditions if it

can predict outcomes

with

reasonable agreement. Neverthe-

less, this does not necessarily imply calibration in general.

Models should be calibrated over a wide range of operating

conditions so that the modeler can rely on model predic-

tions (Walski 1986). As models are only approximations of

the actual systems

that are

being represen ted, the reliability

of model predictions depends on how weU the model struc-

ture is deflned and how well the model is parameterized

(Hogue et al 2006).

A critical step in m odel calibration relates to the quality

and density of observation data which may contain

significant measurement errors. Even small measurement

errors can lead to large errors in estimated parameters


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72 M Mndez eta l Automated parameter optimization of a water distr ibution system

Journal of Hydroinformatics | 15 1 { 2013

(Goegebeur Pauwels 2007). As observation data accuracy

migbt be in tbe same order of magnitude of tbe measure-

ment errors, including this kind of data in tbe calibration

process will most likely produce misleading results. Only

accurate observation data sbould be used for calibration.

Walski

et al

(2004) suggest tbat very high quality field

data (e.g. pressure and elevation data) would be accurate

to 0.1 m. Good quality field data are considered accurate

to m. Model predictions would significantly deteriorate

wb en field data are accurate to 3 m, whicb reflects po orly

collected data. On the other hand, sufficient observation

data may be infrequent an d only collected at select locations

(Kang Lansey 2009). Tbis is particularly true in develop-

ing countries wbere severe limitations constrain the scope

and success of public sector projects, particularly water

supply and water quality (Lee Scbwab 2005).

Calibration by param eter optim ization is a bigbly under-

determined problem as the number of unlaiowns is greater

than the number of observations (Walski

et al

2006). In a

real system, tbere can be hundreds of unlmowns and only

a relatively small number of observations. As tbe number

of unknowns greatly exceeds the number of observations,

tbere is little confidence in the model predictions. Tbere

can be too many solutions tbat yield equally good results

in terms of a predefined objective function witbin the

model's parameter space (Duan

et al

1992). Tbis situation

has resulted in tbe development of tbe concept of 'equifinal-

ity' which recognizes that alternative sets of control

parameters within a model are capable of producing reason-

able estimates of the system response as measured by

objective functions defining tbe goodness of fit (Beven

Binley 1992; Fang Ball 2007). Tbis ambiguity bas serious

impacts on parameter and predictive uncertainty and conse-

quently limits tbe applicability of a model. To reduce tbe

number of unlcnowns and improve reliability on model pre-

dictions, the parameter space must be constrained (Walski

et al 2004).

An obvious constraint for any model would be grouping

param eters into classes of similar cbaracteristics (e.g. groups

of same internal diameter, m aterial, demand sector, etc.). If

all parameters witbin a model are open to adjustment,

mo delers m igbt end u p d oing wb at Walsld (1986) calls 'Cali-

bration by compensating errors'. An error in tbe estimated

roughness can be compensated for by an error in demand.

Compensating errors can take over tbe calibration process

and produce numerically correct, but pbysically meaning-

less, solutions. Under these circumstances, a model would

be matcbing observations rather than determining the sys-

tem's optimal parameters. On the other hand, only those

parameters sensitive to field observations sbould be

included in the calibration p rocess. If insensitive para meters

are included, the parameter space would increase and

model predictions will have littie chance of improving tbe

goo dne ss of fit (Zagblou l Abu Iefa 2001).

Model calibration b as traditionally been a trial-and-error

process. In tbis approach, values of selected parameters are

individually adjusted in a systematic manner until corre-

lation between observed and modeled values no longer

improves. Calibration by trial-and-error is a time-consuming

and difficult task, as tbe large number of potential

unlinowns makes it impossible to analytically solve all cali-

bration parameters (Walski et al 2006). Wben a trial-and-

error optimization approacb is followed, tbe goodness of

fit of tbe optimized model is essentially based on tbe mode-

ler's judgmen ts and experience (Ibrabim Liong 1992; Kbu

et al

2006). Since the judgment involved is subjective, it is

difficult to explicitiy assess tbe confidence of tbe model pre-

dictions (Kumar et al 2009). As deviations between

observed and modeled values migbt be outside tbe accepta-

ble tolerance range, a mucb more 'tuned' optimization may

be required. In tbis case an automated optimization

approacb may be used.

In an automated optimization app roacb, parameters are

adjusted automatically according to a specified search

scheme and quantitative numerical measures of tbe good-

ness of fit (Skahill Dob erty 2006). Tbe development of

automated optimization procedures bas mainly focused on

using a series of objective functions to evaluate the goodness

of fit of the o ptimized m odel. Algorithms try to minim ize tbe

deviations between the observed and modeled values in

order to find the global minimum of tbe selected objective

function (Savic

et al

2009). Nevertbeless, as calibration is

a bighly underdetermined problem, it is impossible to

know whether any automated or manual calibration

approach is actually correct. This situation migbt worsen if

an automated calibration approach is followed, as cali-

bration algorithms might contain little knowledge of the

physical laws that govern a model's structure.


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Several automated techniques have been used to cali-

brate WDS models including genefic algorithms (Jamasb

et

al

2oo8; Nicklow et

al

2009; Kovalenkoe t

al

2010), B aye-

sian-type procedures (Kapelan et al 2007) and Markov

Chain Monte Carlo simulations (Peng

et al

2009). V arious

calibration tools have been incorporated into commercially

available software packages including WaterGEMS's

Darwin-Calibrator (Wu

et al

2002; Walski

et al

2006),

H2ONET Calibrator (Wu et al 2000) and InfoWafer Cali-

brator (de Schaetzen et al 2010). Although these

applications are robust and well proven, little attention has

been paid to end-user applications and product development

(Abe & C heung 2010). This is particularly true in the context

of free and open source software applicafions. Water ufilifies

in developing countries might not have access to commer-

cial packages due to budget restricfions. Still the use and

proper calibrafion of WDS models is a necessity.

The parameter estimator package PEST; an acronym for

Parameter ESTimation (Doherty 2005) might eventually be

used to calibrate a WDS model such as EPANET (Rossman

2000).

PEST offers several strategic advantages to m odelers.

Firstly, PEST is a model-independent application. This

avoids changes to the original code as it communicates

with a model through its own input and output files. Sec-

ondly, PEST has successfully been used to calibrate

numerous types of models including, groundwater models

(Doherty 2005; Christensen & Doherty 2008), hydroiogicai

models (Arabi

et al

2007; Immerzeel & Droogers

2008;

Bah-

remand & de Smedt 2010), soil layer and soil moisture

analysis (Tischler

et al

2007) and hydraulic models

(Koppel & Vassiljev 2009; Maslia

et al

2009). Thirdly,

PEST can be used to carry out various predictive and

exploratory tasks including sensitivity, correlation and

uncertainty analysis. Fourthly, PEST is freely available fo

the public and is user-oriented. As modeling of WDSs is

just emerging in developing regions, particularly in small

water supply systems (municipalifies and rural commu-

nifies), it would be of great importance to explore the

possibility of using available free and open source software

applications in this context. This would gradually promote

a 'modeling culture' which may generate demand for more

comprehensive, commercially available tools.

The aim of this study is to determine if model indepen-

dent parameter estimator PEST can be used to develop a

fully calibrated extended-period model for a WDS using

the public domain model EPANET.

METHODOLOGY

La Sirena's WDS was used as a case study to examine the

advantages of using a fully automated optimization

approach versus a trial-and-error opfimization. La Sirena is

a rural community located in the department of Valle del

Cauca, Municipality of Cali, Colombia (Figure 1).

The system is served by a single mulfi-stage filtrafion

(MSF) water treatment plant which produces approximately

0.012 m'' s ^ and consists of small diameter PVC pipes, ran-

ging from 20 to 85 mm (Table 1). The system contains four

ground WSTs that create four different pressure zones

(Figure 1). Each WST is equipped with a Float Actuated

Valve (FAV) that co ntrols flow admission an d p revents over-

flow. Isolation valves prevent flows among pressure zones.

A comp lete topographic survey of the system was performed

using a total station which allowed elevation data to be accu-

rate to 0.02 m.

Since the system co nsists mainly of small diameter pipes

and minor losses could have a significant impact, the

number of fitfings for each individual pipe was carefully

identified. The types of fittings included tees, elbows,

bends, contracfions, expansions and fully open isolafion

valves. Mino r loss coefficients (K) we re assigned to thes e fit-

tings based on typical values found in the literature (Mays

2000

Observation data for the calibration process included

water levels in the four storage tardes and nodal pressures

in seven nodes (Figure 1). A period of 7 consecutive days

(168 hr) was used for observafions recording. Observafions

were taken during normal operafing conditions and there-

fore,

considered highly representative. Pressure gauges

used to record nodal pressures were accurate to

m while

pressure sensors at WSTs were accurate to 0.1 m. The fime

interval of water level and nodal pressure observafions

was 60 min, sparsely distributed through out the enfire

recording period. In most cases, these observations rep-

resent discrete values recorded at the top of the hour. As

recommended by Walski

et al

(2001), when possible, nodal

pressure observafions were performed at maximum head


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74 M. Mndez eta i Automated parameter optimization of a water distribution system

Journal of Hydroin fortnatics | 15 1

2013

/

y il

^

>v II

\

y

Legend:

isolation Vaiv es

Monitoring Node s

A Wate r Storage

Tanks

Pressure Zone 1

i Pressure Zone 2

1 Pressure Zone 3

, I

i ; Pressure Zone 4

0 75 150 225 300

1 , 1 , 1

Scaie in Meters

Figure 1 I La Sirena s water distribution s ystem mod el.

Tabie 1 I Town of La sirena s water d istribution system characteristics

C o m p o n e n t Quanti ty

Number of nodes

Number of pipelines

Float Actuated Valves (FAVs)

Internal pipe diameter range (mm)

Total pipeline lengtb (m)

Water storage tank, elevation ( MASL)

and capacity (m'^)

Tank 1

Tank 2

Tank 3

Tank 4

325

27 9

5

20-85

9,554

1 178;56

1,113; 79

1 112;200

1 097;54

Meters above sea level.

loss,

which also corresponded to the system's highest water

demand.

At the time the field data were collected, there was a

total of 851 consumers registered in the Water Utility

records. Each consumer was supplied with a flow meter

which was read once a month. This information was used

to estimate and allocate water demand per consumer.

A period of 12 months between 2009 and 2010 was ana-

lyzed for this purpose. Nearly 96% of water demand was

classified as residential. The remaining 4% was classified

as commercial and/or industrial. A point-based method

was used for nodal demand allocation. This required a

detailed field survey to determine the precise number of

households that were connected to the WDS.

As actual

flow

meter locations were k now n, a spatial func-

tion was used to assign each meter to the closest demand node

(Cabreraet l 1996). To include u nacc oun ted for water, field

measurements of production and metered consumption were

recorded and compared during a period of 24 hr and then

added proportionally to nodal demands. Estimations of leak-

age for each pressure zone were prepared from night flow

and water level measurements. Nevertheless, severe inconsis-

tencies regarding consum ption records an d tmaccounted-for-

water estimations were detected. Therefore, nodal demand

exhibits the highest input data uncertainty for this case study.

This situation has been identified in the literature

(Lansey & Basnet 1991; Kenw ard & H ow ard 1999; Zho u

et l

2000) and solutions such as on-line implementation

of Supervisory Control and Data Acquisition (SCADA) sys-

tems have been suggested to improve knowledge of water


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75 M Mndez et

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Automated parameter optimization of a water distr ibution system

Journai of Hydroinformatics | 15 1 | 20 13

dem and temp oral variation (Kang Lansey 2009; Machell

et al.

2010). However, SCADA systems are beyond the

reach of most water utilities in developing countries. In

recent years, researchers have identified the necessity to

include uncertainty in the calibration of WDS models

including pipe roughness coefficients and nodal demands

(Alvisi Fran chini 2010; Giustolisi Berard i 2on ). Uncer-

tainty quantification of model parameters such as nodal

demands implies a profound knowledge of the system and

the availability of sufficiently long time series data. Again,

this might not be the case for many water utilities.

Demand patterns for each WST and their respective

pressure zones were constructed by analyzing diurnal

dynamics of water demand. These temporal variations

were measured for various consumers of each pressure

zone. The effect of the day of the week on the demand pat-

tern s wa s also analyzed . Since significant differences

between weekday and weekend consumption w ere detected,

two 24-hr diurnal demand patterns (with a temporal resol-

ution of

hr) were constructed for each pressure zone;

one known as weekday demand pattern (Monday to

Friday) and the other known as weekend demand pattern

(Saturday and Sunday). In total, eight 24-hr demand patterns

for the entire system w ere con structed. As this project faced

budget limitations, it was not possible to obtain an indepen-

dent set of data that could have been used for model

validation.

Parameter o ptimization

Calibration and sensitivity analysis of EPANET were per-

formed using PEST, a nonlinear parameter estimation and

optimization package, which offers model independent

optimization routines (Doherty 2005). PEST is based on

the Gauss-Marquardt-Levenberg algorithm (GML) which

searches for the optimum values of the model parameters

by minimizing the deviations between field measurements

(observations) and modeled values (predictions). PEST

uses the sum of the squared deviations PHI (0) as its objec-

tive function (Skahill 2009) and can be mathematically

expressed as:

(1)

where n equals the total number of observations, O, the

observed value on timestep i. Mi is the modeled value on

timestep i an d w is the relative weight attached to each

observed value.

At the beginning of each iteration timestep, the relation-

ship between model parameters and modeled output values

is evaluated for the current set of parameters. For instance,

the derivatives of all observations with respect to all adjusta-

ble parameters must be calculated. These derivatives are

stored as the elements of a Jacobian matrix used for sensi-

tivity an alysis.

During each iteration timestep, PEST varies each adjus-

table parameter incrementally from its currently estimated

value and re-runs the model. The ratio of modeled output

differences to parameter differences approximates the

derivative. For greater accuracy in derivatives' calculation,

parameter values can be both increased or decreased until

the objective function PHI is minimized. PEST determines

whether additional iterations are required by comparing

the parameter change and objective function improvement

achieved through the current and previous iterations

(Skahill Do herty 2006).

PEST also calculates a composite relative sensitivity for

each adjustable param eter wh ich me asures the sensitivity of

all observations with respect to a relative change in the

value of that specific parameter. The sensitivity of a par-

ameter describes the effect that a variation on that specific

parameter has over the whole model generated values. Sen-

sitivity is a useful indicator for assessing the degree to which

a parameter is capable of estimating the variables on the

basis of the o bservation dataset available for mo del optimiz-

ation (Doherty 2005).

PEST communicates with a model through the

model's own input and output files. For PEST to be able

to take control of the model, the model itself must be

executable from the command line. Hence, the optimiz-

ation of a model can be carried out without the

requirement for the model to undergo any changes in its

original structure.

To couple EPANET with PEST, several input files must

be prepared (Figure 2). In the case of EPANET, the model's

master input file has a suffix 'inp' (Rossman 2000). This

ASCII-file contains all the necessary information to run

EPANET from the command line using the EPANET2D.EXE


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76 M. Mndez et

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Automated parameter optimization of a water distribution system

Journal of Hydroinformatics | 15 1 | 20 13

f

Model liiput:

EPANET AS CII file

MNP)

>

f

Read m odel input and

replace input

parameters on

PPESTtemplate files

*. TPL; MN S)

Define objectives:

Model parameters +

observations series

Model:

EPANET2D.EXE

>

PPEST.exe

Programa modules

control over EPANET

Analysis *. PST)

>

>

>

f

Reads and compare

mode l simulations

againts observed

values *. INP)

Figure 2

I Coupling schematics of EPANET and PEST.

extension. After EPAN ET is run, ASCII and binary files with

the suffixes txt and grd are respectively created. For PEST,

three types of input files must be created; a template files

(with suffix tpl ) wh ich is basically a copy of the inp

input-file, a con trol file (with suffix pst ) w hich con tains

all the PEST S control an d nu meric param eters, and finally

an instruction file (with suffix ins ), which allows PEST to

properly read observations from the E PAN ET s txt

output-files. In the course of the optimization process,

PEST creates several output files which are stored for later

inspection (Doherfy 2005).

In mo st cases, the bulk of PEST s run time is consum ed

in running the model

itself

Therefore, a special version of

PEST lmown as Parallel PEST (PPEST) was used instead.

PPEST does not only have all the functionalities of PEST

but also facilitates a dramatic enhancement in the optimiz-

ation performance by allowing PEST to run in parallel.

This is particularly useful when the amount of adjustable

parameters and the model run-times are large. It also

allows modelers to take full advantage of the most recent

and powerful multi-core processors, significantly reducing

the time needed to achieve convergence.


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77 M Mndez et a/ Automated parameter optimization of a water distribution system

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Model setup

Model optimization for the La Sirena's WDS was achieved

in two steps. In the first step, a trial-and-error approach

was followed. Values of the selected p aram eters were indivi-

dually modified in a systematic manner until correlation

between observations and modeled values no longer

improved. Three groups of parameters were taken into

account for tbis step: demand pattern factors [-], nodal

demands (L/s) and (C-factors) [-].

In the second step, an automated parameter optimiz-

ation process was executed by linking EPANET with

PPEST. Two different scenarios identified as 'constrained'

and 'unconstrained' were establisbed to evaluate tbe effects

of parameter space size and compensating errors over tbe

PPEST calibration process. In botb scenarios, 12 parameter

groups were created and tbey included: demand pattern fac-

tors [-], nodal demands (L/s), local losses [-], (FAVs) [-] and

C-factors [-] (Table 2).

For tbe constrained scenario, a total of 24 adjustable

parameters were declared. Demand pattern factors were

fixed to their original values as sufficient confidence was

obtained during their derivation. Nodal demands were

aggregated into four different demand groups, one for each

pressure zone. The same demand multiplier was assigned

to each node witbin eacb demand group. Tberefore, all

nodes experienced a proportionally equal cbange in

demand witbin each specific pressure zone. Minor loss

Tab ie 2 I Set of control settings used for ppEST

coefficients (K) for fittings were not adjusted either. FAVs

loss coefficients w ere adjusted as significant differences

between manufacturer's curves and field bebavior were

detected. Tbrottle Control Valves (TCVs) were used in

EPANET to simulate FAVs. TCVs are commonly used to

simulate a partially closed valve by adjusting tbe minor

head loss coefficient of the valve (Rossman 2000). C-factors

were classified into groups of tbe same pipe diameter. No

distinction was made regarding the age of tbe pipes as tbe

entire system w as co nstructed of relatively yotmg PVC.

For tbe uncon strained scenario, a total of 723 adjustable

parameters were declared, which accounted for the 12

parameter groups mentioned above. Each parameter within

these groups w as individually adjusted by PPEST. Nevertbe-

less, a unique C-factor was used for all pipes in the system.

Observations in botb scenarios, wbicb included water

levels and no dal pressures, were group ed in five observation

groups, four for WSTs and one for monitoring nodes. Since

observations were of two different types, the objective func-

tion relative weight attached to each observation, varied

according to its relative impo rtance in tbe overall pa rame ter

estimation process. Consequently, greater weigbts were

assigned to water level observations since tbey were con-

sidered more reliable tban tbose of tbe nodal pressures.

Concerning tbe ranges for parameter adjustments, a

30%

variation for demand patterns factors was allowed

for tbe un constrain ed scenario (Table 3). As severe inconsis-

tencies regarding consumption records and unaccounted-

for-water estimations were detected, a 30% and 50%

variation in nodal demands multipliers were assigned to

item

Number of adjustable parameters

Num ber of fixed parameters

Number of tied parameters

Number

of observation groups

Num ber and type of parameter groups

Demand patterns factors

Nodal demands

Minor loss coefficients (Ks)

Float Actuated Valves (FAVs) loss

coefficients

C-factors

Scenario

Constrained

723

0

0

5

8

1

1

1

1

Unconstrained

24

192

5 7

5

8

1

1

1

1

Table 3

Ranges for param eters adjustments

Parametergroup

Demand patterns factors (%)

Nodal demands multiplier (%)

Minor loss coefficients (Ks) (-)

Float Actuated Valves (FAVs)

loss coefficients (-)

C-factors (-)

'Walski e t

al

(2001).

Mays (2000).

Ranges of parameters vaiues per

scenario

Constrained

Not applicable

30%

Not applicable

0-350

130-150='

Unconstrained

30%

50%

0-20''

0-350

130-150


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78 M Mndeztal Automated parameter optimization of a water distr ibution system

Journai of Hydroinformatics | 15.1 | 2013

the constrained and unconstrained scenarios, respectively.

Minor loss coefficients {K ) for pipe fittings were allowed

to vary from 0 to 20 based o n typical values found in the lit-

eratu re (Mays 2000). FAVs loss coefficients (K) were

allowed to vary from 0 to 350 based on manufacturer's

pressure drop curves and field measurements. Two fiow

tests con ducted in 69 mm PVC showed C-factor values of

130 and 134 correspondingly. For instance, a lower value

of 130 was selected for C-factors in each scenario while a

feasible upper value of 150 was obfained from the literature

(Walskiet al.2001).

Even when observations were recorded every 60 min a

hydraulic timestep of

min was used for the entire 168-hr

simulation period regardless of the optimization approach

(trial-and-error or PPEST). This is due to the fact that

many observations (mainly nodal pressures) were sparsely

and manually recorded throughout the enfire recording

period and no specific timing for field measurements was

possible.

An Intel Core 7-930, 2.80 GHz multi-core processor

with 24 GB of RAM mem ory was used to run PPEST by con-

trolling one master and eight slaves (two for each of the four

cores of the i-7 processor). PPEST was run in esfimation

mode which means that PPEST confinued with the iteration

procedure until the global minimum was found. All other

PPEST numerical parameters were assigned typical values

as noted in the PEST manual (Doherty 2005).

Efficiency of the model opfimizafion process was

assessed according to two indicators (Maslia et al. 2009):

the Pearson correlation coefficient R),

.ti OiMi) - [nM ]

U m) - (Of))] [Er=i

and the root mean square error (RMSE),

2)

2

3)

where n equals the total number of observations, O, is the

observed-value on timestep i, is the average of the

observed values, M,- is the modeled-value on timestep ian d

M is the average of the modeled values.

R

serves as an indicator of the correlation degree

between observed and modeled values. A value of 1 yields

perfect correlation whereas a value of 0 indicates that data

are uncorrelated. The RMSE provides information on the

average error between observed and modeled values.

RESULTS AN D DISCUSSION

RMSE and R for modeled values of water levels and nodal

pressures were obtained from the two opfimization

approaches: trial-and-error and PPEST. Overall, both PPEST

optimization scenarios resulted in lower RMSE and higher R

than those ob tained by trial-and-error. Nevertheless, little vari-

ation in RMSE andRbetween PPES T scenarios can be seen

(Table 4).

This is more evident for water levels in WSTs as RMSE

remains within the 0.1 m accuracy expected for pressure

sensors. In the trial-and-error approach, RMSE for water

levels remains over the 0.1 m accuracy except for WST 2

(0.06 m). Sfill, RMSE values obtained by trial-and-error are

considered acceptable if limitations in observation data

acquisition are taken into account.

WSTs 1, 3 and 4 experienced the highest RMSE

reductions, 54.8, 31.2 and 32.8%, respectively for the

PPEST constrained scenario. Very similar values (57.6,

41.4 and 36.7%) were obtained for the PPEST uncon-

strained scenario. WST 2 is the excepfion for both PPEST

scenarios since its RMSE increased rather than decreased

with respect to the trial-and-error approach.

As mentioned, PPEST attempts to globally m inimize the

objective function PHI taking into consideration the contri-

bufions of the available observation groups. Since the

relative weight of the water level observation group was

the same for all tanks, it appears that PPEST allowed for

higher deviations in WST 2 in order to compensate for

errors in the remaining WSTs and pressure monitoring

nodes. Maslia et al. (2009) obtained similar results in their

EPANET-PEST analysis of the Holcomb Boulevard WDS,

with RMSE reducfions between 26.5 and 91.4%. Their

study included four WSTs, one of which experienced an

increase in RMSE after fhe optimization with PEST.

In our case, correlation coefficients increased in both

PPEST scenarios, with WST 1 presenfing the highest R


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Journaiof ydroinformatics | 15 1 | 2013

Tab ie4 RMSE

and i?

modeled values

of

water levels

and

nodal pressures

for the

town

of La

Sirena s wa ter distribution system model

Optimization approach

Component

Water levelsinTank1

Water levels

in

Tank

2

Water levels

in

Tank

3

Water levelsinTank4

Nodal pressures

Triai-and-error

Objective function

RMSE (m)

0.134

0.064

0.116

0.158

4.678

R(-)

0.821

0.895

0.946

0.879

0.981

Constrained PPEST scenario

Objective function

RMSE (m)

0.061

0.068

0.080

0.106

1.852

0.965

0.917

0.982

0.940

0.998

Percentage change

R S m)

-54.781

6.438

-31.194

-32.825

-60.407

R(-)

17.548

2.429

3.733

6.979

1.732

unconstrained PPEST scenario

Objective function

RMSE (m)

0.057

0.073

0.068

0.100

1.545

0.975

0.911

0.987

0.947

0.999

Percentage change

RMSE (m)

-57.617

13.633

-41.437

-36.713

-66.976

18.707

1.796

4.252

7.777

1.776

increase,

17.6 and 18.7 for the

constrained

and

uncon-

strained scenarios, respectively.

In the

case

of

nodal

pressures, PPEST significantly reduced RMSE with respect

to

the

trial-and-error approach

as it

passed from

4.7 to

1.9 m for the

constrained scenario

and 1.5 m for the

uncon-

strained scenario.

In all

cases, RMSE

for

nodal pressures

weis higher than

the m

accuracy expected

for

pressure

gauges.

The

difference

of

roughly

0.4 m in

RMSE between

th e

two

PPEST scenarios might seem trivial,

but it

does

have profound implications

in the

optimization process.

As

the

method used

in

EPANET

to

solve

the now

conti-

nuity

and

headloss equations

is

driven

by

nodal demands

and energy losses (Rossman 2000), small measurement

errors

and

large parameter spaces

can

lead

to

large errors

in estimated parameters. This

is

clearly

the

case

of the

PPEST unconstrained scenario, where every single

par-

ameter

was

declared adjustable

in

PPEST (Table

3).

These

conditions defined

an

unmanageable parameter space

of

723 parameters, which implied

the

optimization

of

every

single nodal demand, dem and pattern

and

minor loss coeffi-

cient

in the

system.

The

conditions stated

for

this scenario

evidently forced PPESTtocalibratethemodelbycompen-

sating

for

errors

as

suggested

by

Walsld (1986). Errors

in

nodal demands were most likely compensated

by

errors

in

demand patterns, minor loss coefficients, C-factors

and

pretty much

any

other adjustable parameter.

The

problem

is aggravated

by the

fact that nodal demand

is

still

the

most uncertain

and

variable parameter

in

this case study.

The large parameter space

and

considerable compensating

errors resulted

in a 1.5 m

RMSE

for

nodal pressures,

but

also required considerable change

in the

individual nodal

demands.

After

the

PPEST optimization,

all

nodes experienced

some level

of

change

in

demand (Figure

3)

with

an

absolute

average change

of

34.2%. Several nodes were pushed

towards

the

predefined upper

and

lower demand bounds

(Table

3).

Over

40

nodes exhibited

a -50

change

i

demand

and

over

50

nodes experienced

a -f50

change

in

demand.

The

parameter space

was too big for

PPEST

t

search effectively.

Total water demand

for the

system changed only

slightly,

as it

passed from

10.58 to

10.471s ' after

th

PPEST unconstrained optimization. This represents

a per

centage change

in

total water demand

of

around

1 .

This

reaffirms once more

the

presence

of

considerable compen-

sating errors.

In

this respect, PPEST

is

most likely

producing

a

numerically correct

but

physically meaningless

solution. Furthermore, PPEST

is

possibly matching obser-

vations rather than determining

the

system's optimal

parameters

as

there

is an

excessive number

of

parameter

groups

and

insufficient observation data. Baranowski

(2007) encountered similar conditions when calibrating

nodal demands

in two

experimental EPANET networks

using PEST and Newton-Raphson algorithms. Both methods

maximized hydraulic

and

water quality standards

but

also

required significant change

in the

nodal demands. C-factors

had little chance

to

actively participate

in the

optimization

process

as one

unique parameter

was

defined

for all

pipes.

An optimized C-factor value

of 137 was

found

for al

pipes that remain close

to the

C-factor values found

fo

th e

two

fiow tests conducted

in 69 mm PVC (130 and 13

correspondingly).

At the end of the

optimization process,

the percentage change between trial-and-error

and

PPEST

optimized values

was

2.1%

for

local losses

and 8.5 fo


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| Automated parameter optimizationofa water distribution system

JournaiofHydroinformatics | 15.1 | 2013

-50-40 -40-30 -30-20 -20-10 -10-0 0-10 10-20

Difference in nodal demand

Figure3

I

Percent change

for

nodal demand after PPEST unconstrained optimization scenario.

20-30 30-40 40-50

FAVs. These percentages could seem small compared to

those experienced by nodal demands or demand pattern fac-

tors; however,it isrelativetothe overall sensitivityofeach

adjustable parameter.

In the case

of

the PPE ST constrained scenario, only

24

adjustable parameters were declared (Table 3). The remain-

ing parameters were either fixedortiedtoother param eters.

In PEST,a tied parameter represents a parameter thatis

linkedto amaster parameter.Inthis case, only the master

parameteris optimized andthe tied parametersaresimply

varied with this parameter, maintaining a constant ratio,

through the calibration process.

This representsamuch smaller parameter space. Nodal

demands were aggregated into four different demand

groups, one

for

each pressure zo ne. Since demand pattern

and minor loss coefficients were fixed,thesolution of the

parameter space was limited tofour noda l dem ands (four

parameters), four FAV valves 12parameters) and eight

C-factors (eight parameters). In this case, the absolute

average changeofnodal demand w as 13.4 , much lower

than the 34.2 found for the unconstrained scenario.

None of thenodes w ere pushed towards the predefined

upper and lower demand bounds. This would appearto be

the resultof compensating errors from theC-factors group

(Table 5).

Table5

I

Optimized C-fac tor vaiues after PPEST constra ined optimiz ation sc enario

Pipe internal diameter mm) initial c-factor

PEST optimized C-factor

84

69

57

45

40

31

25

19

36

36

36

36

36

36

36

36

36

37

36

4

4

36

4

Nevertheless, almost all diameter groups remainedin

the lower bound

of

130. Only pipes with internal diameters

of 69 and 45 mm w ere moved to higher values (138 and 150,

respectively). Since demand patterns and minor loss coeffi-

cients were fixed, compensating errors for this scenario

are considerably lower than those of the unconstrained

scenario. For instance, this solutionis more physically cor-

rect asPPESThad a higher chance to findtheoptimum

setofparam eter v alues. Still, no validation data exist to sus-

tain this statement. Maslia et al (2009) experienced

comparable resultsintheir analysisofHolcomb Boulevard

WD S as PVC C-factor change from its original 145 proposed


11/16

81 M. Mndez et

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Autom ated parameter optimization of a water distribution system


value to an optimized range of 147-151 optimized values.

This is more likely to be a consequence of the inherent

smoothness of plastic materials and low relative sensibility

as compared to other materials such as cast iron or steel

which are greatly affected by age, corrosion and deposition

processes (Koppel & Vassiljev 2009).

Total system water demand also changed only slightly,

as it passed from 10.58 to 10.50 ls ^ for th e PPE ST con-

strained scenario. Regarding temporal variation of water

levels at WSTs, a closer match between observed and mod-

eled values can be appreciated for the PPEST scenarios

(Figure 4). This is particularly true for WSTs 1 and 3

(Figure 4(a),(c)), whereas little difference can be seen for

tank s 2 and 4 (Figure 4(b),(d)) regardless of the optimization

approach. The PPEST unconstrained scenario was partly

based on the adjustment of the demand pattem factors

(Table 3).

PPEST optimized weekday demand factors are gener-

ally in good agreement with those initially obtained by

analyzing diurnal dynamics of water demand, particularly

for WSTs 1 and 3 (Figure 5(a),(c)).

There are, however, local differences that show a

larger variation around the hours of higher water

demand , betw een 0900 and 1400 hr, p rincipally for

WSTs 2 and 4 (Figure 5(b),(d)). This could be attributed

to the relative sensitivity of those pattern factors. None-

theless, little confidence can be attributed to these

patterns as compensating errors are very high for the

PPEST unconstrained scenario.

Parameter sensitivity

As calculated by PPEST, the WDS model is mostly

sensitive to nodal demands since this group of

parameters exhibits the highest average composite sensi-

tivities for both constrained and unconstrained

scenarios, 2.86 and 2. 2 6 x lO \ respectively (Table 6).

As discussed above, EPANET is controlled by water

12 24 36 48 6

0.5

12

24 36

Time hrs)

48 60

12 24 36 48

Time hrs)

60 72

EXPLANATION Obs ervations - PPEST Con strained PPEST Unco nstraine d Trial-and-Error

Figure 4

I PPEST optim ized wa ter-lev el values for a time lapse of 72-hr.


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82

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JournalofHydroinformatics | 15.1 | 2013

10 12 14 16 18 20 22 24

Time iirs)

6 8

1 12 14

Time iirs)

16 18 20 22 24

EXPLANATION

Initial Estimation

PPEST Unconstrained

igure

5I

Initially estimated versus PPEST unconstrained optimized weei


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83 M Mndez eta l | Automated parameter optimization of a water distribution system

Journai of Hydroinformatics | 15 1 | 2013

are only determinant if sufficient head loss is experienced

across the WDS. This is clearly not our case, as velocities

and head losses are quite small most of the fime.

Objective function

From a practical standpoint, it is essenfial to assess the

number of model calls that PPEST required to minimize

its objective funcfion PHI {0 . For the unconstrained scen-

ario,

PPEST needed to run EPANET 6507 fimes in order

to find what PPEST considers to be the global minimum .

This global min imu m w as found after the fifth iterafion,

achieving a PHI reduction of nearly 77%, as its initial value

was reduced from 7.77 to 1.817 m^. As stated by G oegebeur

& Pauwels (2007), even when PHI differs from RMSE, the

minimization of this function also leads to a minimization

of the RMSE (Table 4).

After running in estimafion-mode, PPEST executed four

addifional iterafions after the global minimum was found,

just to ensure that it did not encounter a local minimum.

A total of 10,845 EPANET calls were needed to complete

the optimization process, wh ich lasted for 13.2 hr, clearly

indicafing a high computational burden. If PEST instead

of PPEST had been used in this case, an estimated 90.4 hr

run would have been needed to accomplish the same

result. This is quite predictable as PEST's final report

states how long it needs to complete one EPANET's run.

For the m ore realistic and physically correct co nstrained

scenario, PPEST com pleted the who le opfimizafion process

in 0.74 hr, almost 18 times faster th an the unco nstrained

scenario and with similar outcomes as PHI was reduced

from its original trial-and-error value of 7.77-2.065 m^.

The most fime consuming part of PPEST's operafions is

related to the calculation of the Jacobian matrix through par-

tial derivatives, which requires two model calls for each

adjustable parameter. Nonetheless, the calculation of the

Jacobian matrix permits PPEST to derive important by-

products such as relative and absolute sensitivity, corre-

lation, uncertainty and Eigenvectors which are of great

importance in post-optimization analysis.

On the other hand, PPEST is capable of fixing par-

ameters based on a given sensitivity threshold. This could

help improve PPEST's performance through modeler inter-

venfion (Doherty 2005). Finally, it is important to note that

EPANET is a fully distributed complex hydraulic solver

which demands a great deal of computafional resources by

its lf

This clearly increases the time lapse needed for

PPEST to achieve convergence.

CONCLUSIONS

The feasibility of using the model independent parameter

esfimator PPE ST to develop a fully-calibrated extend ed

period model for a WDS using EPANET was investigated.

It was found that the results from a fully automated cali-

brafion technique like PPEST should be used with caufion.

As calibration is a highly underdetermined problem,

PPEST may produce numerically correct but physically

meaningless solutions if insufficient restrictions are applied.

The two PPEST scenarios analyzed in this study further

emphasize this situation.

The uncon strained scenario show ed that if all parameters

are open to adjustment, considerable com pensating errors are

introduced into the optimization process. Under these cir-

cumstances, PPEST was capable of matching observations

but incapable of determining the opfimtim set of param eters

for the system. As the number of unknown s was mu ch greater

than the number of observafions, the parameter space was

simply too large for PPEST to find a proper solufion. It was

demonstrated that including insensitive parameters signifi-

cantly deteriorates model's solufions and imposed a heavy

computational burden on the whole optimization process.

On the other hand, the constrained scenario represents

a more properly discretized and therefore, more reliable

scheme as parameters were grouped in classes of similar

characteristics and insensitive parameters were fixed. This

had a profound impact on the parameter space as adjustable

parameters were reduced from 723 in the unconstrained

scenario to 24 in the constrained scenario. In this regard,

the computational requirements were much lower as the

whole opfimization process took 0.74 hr as com pared to

the 13.2 hr needed for the uncons trained scenario.

The model was mainly sensitive to nodal demands as

average composite sensitivities for both constrained and

unco nstrained PPEST scenarios reache d 2.86 and 2.26 x

10 \ respectively. The FAVs loss coefficients r epre sente d

the second most sensitive group of parameters as their


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8 M Mndez et

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| Automated parameter optimizationof awater distr ibution system

Journai

of

Hydroinformatics

| 15 1 | 2013

average composite sensitivity reached values of

1.8080

x

10 ^ and 2.5980 x 10 for the constrained and uncon-

strained PEST scenarios, respectively. The remaining

groups of parameters, including C-factors, showed very low

sensitivities, two to three orders in magnitude lower than

that of nodal demands. A considerable RMSE reduction

was obtained for the constrained scenario, particularly for

nodal pressures as its value decreased from 4.68 to 1.85 m.

The constrained solution, even when it is valid only for

the system's normal operating conditions, clearly demon-

strates that PPEST has the potential to be used in the

calibration of WDS models.

FUTURE WORK

Further investigation is needed to determine PPEST's per-

formance in complex WDS models over a wide range of

operating conditions. This may include the evaluation of

control valves, pump curves, control rules and water quality

issues.

An EPANET-PEST utility software could be devel-

oped to facilitate communication between both packages.

This would allow users to run PEST directly in a graphical

environment, where many of the variables and outcomes

of the calibration process could be viewed and externally

manipulated.

C K N O W L E D G M E N T S

The authors wish to thank Junta Administradora del

Acueducto La Sirena, La comunidad de la Sirena,

Estacin de Investigacin y Transferencia de Tecnologa

de Puerto Mallarino and Cinara-Universidad del Valle for

their help and guidance in the development of this project.

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